Transformation to a Portfolio Model

The proposed stress-testing model could be easily transformed into a portfolio model (the model dedicated to the estimation of unexpected losses). In the case of a portfolio model, a macro-forecast should be excluded from the model by replacing

the forecasted Mqi values by the random values Mqi. The distribution function of

Mqi could be calibrated using the historical values of macro-variables.

One of the most flexible approaches that could capture the time evolution of macro-variables is the ARIMA model. The ARIMA model would capture the following aspects of time evolution of macro-variables:

(c) Deviations from trends (prior to the period, error affects the current period’s errors).

2. The random component—normally distributed random variables with a zero mean and covariance matrix (estimated on the basis of historical deviations of the real values of the macro factors from the ARIMA model).

In the case of the portfolio model, the Monte Carlo simulation schema should be modified in the following way:

1. Using the ARIMA model, Mqi values are generated for the estimation period.

2. The MC algorithm for the stress-testing model is started, in which, instead of

forecasted macro-variables Mqi, random variables Mqi are used.

Conclusion

The proposed model meets all of the requirements mentioned in this article’s introduction. The model could produce estimates both of losses due to borrowers’ defaults and changes in the rating structure. The model is based on the functional dependence between dynamics of macro-variables and defaults; therefore it could be calculated for baseline and stress-scenarios. Comparison between the results in different scenarios will give us estimates of the changes of direct losses (defaults) and RWA changes (rating structure) due to stress events. The model could also be easily extended to the credit VAR model; therefore a bank could make consistent comparisons between stress-testing results and unexpected losses.

• Tails are fitted separately with a Pareto distribution. It is a base distribution from an extreme value theory in the sense that every distribution of any extreme value can be transformed into a Pareto distribution. It has the form:

• pc (t) = p (ti*), where ti_1* < t < ti* is the time between the two subsequent transactions i — 1* and i *.

• Apc (t, At) = pc (ti*+n*) — pc (ti*), is the price change observed sampling the time series pc (t) at a time scale At. This change is caused by n* consecutive trades. The number of trades n* is a stochastic variable for each fixed value of the time scale At.

We develop an algorithm that samples the time series of trade time ti in order to determine the index i* and n* of trades that we need to observe the series in continuous time pc (t), Apc and rc. We report in Table 1 some sample statistic about log-returns corresponding to the smallest time scales studied in this work. When we observe prices in continuous time, the empirical returns distribution is more fat­tailed than that defined in trade time. The increase of kurtosis can be explained if we think to price process as a subordinated random process. We give same details on the subordination hypothesis in Sect. 3.2.

[2]See, for example, phase trajectories of several well-known simple chaotic systems, such as the Mackey-Glass and Genesio-Tesi systems, and trajectories of purely chaotic system such as Wiener process.

[3] Notice that for returns the discretization effect is different from clustering: discretization is a

consequence of the fact that price is defined on a grid, while clustering denotes the preference for some price variations over others.

[7] For the GM model, we simulated straight incorporation of inventory costs into the bid-ask spread and this leads to explosive growth of spread and price. Program code and results are available upon the readers’ request.

[8]We made our simulation using a computer program, which is written in R. We can provide the program code upon the reader’s request. You can send your request to the author via e-mail. We have a code to simulate the GM model for straight incorporation of inventory risk into bid-ask spread, and the first and the second stages of modification.

[9] We must notice that typically stochastic volatility models are defined not within the framework of Eq. (2), but as an extension of stochastic differential equation of the geometric Brownian motion. Strictly speaking, these equations do not always have solution in form of (2).

[14]Also noteworthy is that some Black Swans may be Dragon Kings to those with special insight: astronomers might forecast an asteroid impact, security analysts might uncover a high likelihood of a terrorist attack, while safety engineers might have insight about escalating risks of an industrial breakdown.

[15]I can’t help but think that we see this same effect in the climate change discussion, with climate scientists as Innovators at the periphery of the public network, struggling to cross the chasm of global adoption.

[16]All VaR backtesting is based on the standard RiskMetrics methodology (exponential weighting with 0.94 decay).

[19]As measured by one day standard deviation residual, using dynamic RiskMetrics volatility estimation.

[20]Impressively, October 2007 was the bubble peak forecasted by Didier Sornette’s LPPL models.

[21]Interestingly implied volatility did spike in THB options prior to the devaluation as an early warning signal, as documented by Malz (2011).

[22]This build-up of hidden risk until a dramatic collapse is a common theme with pegged currencies: Argentina experienced a similar.

[23]I am reminded of a statement by a HK hedge fund manager about Goldman Sachs, after we discussed their use of VaR outlier signals to exit subprime. “They’re like geologists who make their living right top of all the world’s fault lines line, monitoring every tremor.”

[25]A detailed white paper of HeavyTails is available upon request, and visit HeavyTails. com for more information.

Innovator would consider all four quadrants (and more perspectives) in their risk assessment.

[27]A famous example is legendary investor George Soros who developed gut instincts about risk. He was known to presciently exit positions by listening to his body’s stress signals.

[28]Dr. Atul Gawande’s “Checklist Manifesto” (2009) provides great insights about importance of well designed checklists for managing risk, with case studies from medicine, aviation, investments, and construction.

[29]This viewpoint is consistent with Sharpe’s (1992) decomposition of a mutual fund’s return into two components: the “style” (i. e. asset-class factors, such as large-cap stocks, growth stocks etc.) and “selection” (i. e. an uncorrelated residual).

[30]The “cleaned” P&L is calculated in the same way as the “dirty” P&L, but without taking into account position changes during the VaR horizon. Paid and received fees and commissions are omitted from the calculation.

[33]A typical hierarchy within a trading function includes trading books run by trading desks, which, in turn, are operated by individual traders. The Basel Committee on Banking Supervision has recently attempted to give a regulatory definition of a trading desk (Basel Committee 2013).

[34]“The risk measurement system should be used in conjunction with internal trading and exposure limits. In this regard, trading limits should be related to the bank’s risk measurement model in a manner that is consistent over time and that is well understood by both traders and senior management.” (Basel Committee on Banking Supervision 2006, §718(Lxxiv)-f).

[36]According to StraBberger (2002), this approach can be extended to a stock portfolio. In our following discussion, we consider a single position in a stock.

[37]These are the mean and the standard deviation of portfolio returns, as VaR is calculated using the variance-covariance approach.

[38]Delta for a portfolio with long and short positions is calculated for changes in daily position limits and not in the stock prices.

[39] Several research firms provide estimates of HFT activity for subscribers; examples are the TABB Group, the Aite Group, and Celent. Publicly, this information is available in articles such as “The fast and the furious”, Feb. 25, 2012, The Economist and “Superfast traders feel the heat as bourses act”, Mar. 6, 2012, Financial Times.

[41]Prior to the incident, there had been a change in the storm-time variation index Dst (a space weather indicator) by 43 nT over a period from 4.00 p. m. (local time) on December 19, 2005 to 6.00 a. m. on December 20, 2005. Although this was, of course, only an increment of variation rather than an absolute value, one should keep in mind that a variation level of —50 nT is equivalent to a mild storm rated on the National Oceanic and Atmospheric Administration (NOAA) geomagnetic storm scales as a G1 (such geomagnetic storms sometimes affect the start of animal migration, cause fluctuations in electric power systems, etc.).